Answer to Question #351046 in Statistics and Probability for Bob

Question

2. QUESTION:

A fair coin is tossed, and a fair die is thrown. Write down sample spaces for

(a) the toss of the coin;

(b) the throw of the die;

(c) the combination of these experiments.

Let A be the event that a head is tossed, and B be the event that an odd number is thrown. Directly from the sample space, calculate P(A ∩ B) and P(A ∪ B).

Answer 1

(a) $\{HEAD,TAIL\}$

(b) $\{1,2,3,4,5,6\}$

(c)

$\{(HEAD,1), (HEAD,2), (HEAD,3),\\ (HEAD,4), (HEAD,5), (HEAD,6),\\ (TAIL,1), (TAIL,2), (TAIL,3),\\ (TAIL,4), (TAIL,5), (TAIL,6)\}$

The sample space of combination of the experiments has $12$ elements with equal probabilities $\frac{1}{12}.$

$A\cap B=\{(HEAD,1), (HEAD,3), (HEAD,5)\}$,

$|A\cap B|=3$, hence $P(A\cap B)=3\cdot \frac{1}{12}=\frac{1}{4}.$

$A\cup B=\{(HEAD,1), (HEAD,2), (HEAD,3),\\ (HEAD,4), (HEAD,5), (HEAD,6),\\ (TAIL,1), (TAIL,3), (TAIL,5)\},$

$|A\cup B|=9,$ hence $P(A\cup B)=9\cdot \frac{1}{12}=\frac{3}{4}.$

Answer: $P(A\cap B)=\frac{1}{4}, P(A\cup B)=3\cdot \frac{1}{12}=\frac{3}{4}.$